# Prove Invertibility of Matrix I + A for Projection Matrices A

• danik_ejik
In summary, the conversation discusses how to prove that if A is a projection matrix with A^2=A, then I+A is invertible and its inverse is I-A/2. The participants suggest using a series expansion and show that the product of I+A and I-A/2 equals I, which proves the statement. They also mention the definition of 'inverse'.
danik_ejik
Hello,
I need a hint on how to begin this proof, please.

Prove that if A is a projection matrix, A2=A, then I + A is invertible and
(I + A) -1 = I - $$\frac{1}{2}$$A.

I would write B=a*I+b*A and compute B*(I+A) and (I+A)B and see for what values of a and b I find an inverse. Note that A^2=A, so there are no higher powers in the series expansion of the inverse.

If I+A and I-A/2 are supposed to be inverses then their product should be I, right? Is it?

Well, $$I-A/2$$ is a matrix. Why don't you try to see what you get if you multiply it by $$I+A$$? Maybe it will tell you something?

I tried multiplying (I + A) and (I - A/2) and indeed it equals I. That's it!? That proves it ?

danik_ejik said:
I tried multiplying (I + A) and (I - A/2) and indeed it equals I. That's it!? That proves it ?

Sure that proves it. If MN=I then M=N^(-1) and N=M^(-1). It's the definition of 'inverse'.

thank you

## 1. How can you prove the invertibility of a matrix I + A for projection matrices A?

To prove the invertibility of a matrix I + A, you can use the fact that for a projection matrix A, (I + A) is also a projection matrix. This means that (I + A)^2 = (I + A), which implies that (I + A) is its own inverse. Therefore, (I + A) is invertible and its inverse is (I + A) itself.

## 2. What is a projection matrix and what properties does it have?

A projection matrix is a square matrix that, when multiplied by itself, gives the same matrix. This means that it projects a vector onto a subspace, hence the name "projection" matrix. It has the properties of being symmetric, idempotent, and having eigenvalues of 0 or 1.

## 3. Why is it important to prove the invertibility of a matrix I + A for projection matrices A?

Proving the invertibility of a matrix I + A for projection matrices A is important because it shows that the matrix is well-defined and has a unique solution. It also allows for easier calculations and simplification of equations when working with projection matrices.

## 4. Can a projection matrix be singular?

Yes, a projection matrix can be singular. If a projection matrix has eigenvalues of 0 or 1, it will be singular.

## 5. How does the invertibility of a matrix I + A for projection matrices A relate to linear transformations?

The invertibility of a matrix I + A for projection matrices A is related to linear transformations because projection matrices represent linear transformations that project vectors onto a subspace. An invertible projection matrix ensures that there is a unique transformation from the subspace back to the original vector space.

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